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Dagstuhl Seminar Proceedings, Volume 4461

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  • published at: 2005-08-10
  • Publisher: Schloss Dagstuhl – Leibniz-Zentrum für Informatik

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Document
DOI: 10.4230/DagSemProc.04461.1
04461 Abstracts Collection – Practical Approaches to Multi-Objective Optimization

Authors: Jürgen Branke, Deb Kalyanmoy, Kaisa Miettinen, and Ralph E. Steuer

Abstract
From 07.11.04 to 12.11.04, the Dagstuhl Seminar 04461 ``Practical Approaches to Multi-Objective Optimization'' was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available.
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Jürgen Branke, Deb Kalyanmoy, Kaisa Miettinen, and Ralph E. Steuer. 04461 Abstracts Collection – Practical Approaches to Multi-Objective Optimization. In Practical Approaches to Multi-Objective Optimization. Dagstuhl Seminar Proceedings, Volume 4461, pp. 1-17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{branke_et_al:DagSemProc.04461.1,
  author =	{Branke, J\"{u}rgen and Kalyanmoy, Deb and Miettinen, Kaisa and Steuer, Ralph E.},
  title =	{{04461 Abstracts Collection – Practical Approaches to Multi-Objective Optimization}},
  booktitle =	{Practical Approaches to Multi-Objective Optimization},
  pages =	{1--17},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2005},
  volume =	{4461},
  editor =	{J\"{u}rgen Branke and Kalyanmoy Deb and Kaisa Miettinen and Ralph E. Steuer},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.04461.1},
  URN =		{urn:nbn:de:0030-drops-2551},
  doi =		{10.4230/DagSemProc.04461.1},
  annote =	{Keywords: Multi-objective optimization, evolutionary algorithms, decision support system}
}
Document
DOI: 10.4230/DagSemProc.04461.2
04461 Summary – Practical Approaches to Multi-Criterion Optimization

Authors: Jürgen Branke, Kalyanmoy Deb, Kaisa Miettinen, and Ralph E. Steuer

Abstract
Summary of the Dagstuhl Seminar 04461. Motivation, proceedings, achievements and feedback, future seminars
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Jürgen Branke, Kalyanmoy Deb, Kaisa Miettinen, and Ralph E. Steuer. 04461 Summary – Practical Approaches to Multi-Criterion Optimization. In Practical Approaches to Multi-Objective Optimization. Dagstuhl Seminar Proceedings, Volume 4461, pp. 1-5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{branke_et_al:DagSemProc.04461.2,
  author =	{Branke, J\"{u}rgen and Deb, Kalyanmoy and Miettinen, Kaisa and Steuer, Ralph E.},
  title =	{{04461 Summary – Practical Approaches to Multi-Criterion Optimization}},
  booktitle =	{Practical Approaches to Multi-Objective Optimization},
  pages =	{1--5},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2005},
  volume =	{4461},
  editor =	{J\"{u}rgen Branke and Kalyanmoy Deb and Kaisa Miettinen and Ralph E. Steuer},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.04461.2},
  URN =		{urn:nbn:de:0030-drops-2430},
  doi =		{10.4230/DagSemProc.04461.2},
  annote =	{Keywords: Multi-criterion Optimization, Classical and Evolutionary Approaches}
}
Document
DOI: 10.4230/DagSemProc.04461.3
A New Adaptive Algorithm for Convex Quadratic Multicriteria Optimization

Authors: Jörg Fliege, Christoph Heermann, and Bernd Weyers

Abstract
We present a new adaptive algorithm for convex quadratic multicriteria optimization. The algorithm is able to adaptively refine the approximation to the set of efficient points by way of a warm-start interior-point scalarization approach. Numerical results show that this technique is an order of magnitude faster than a standard method used for this problem.
Cite as

Jörg Fliege, Christoph Heermann, and Bernd Weyers. A New Adaptive Algorithm for Convex Quadratic Multicriteria Optimization. In Practical Approaches to Multi-Objective Optimization. Dagstuhl Seminar Proceedings, Volume 4461, pp. 1-39, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{fliege_et_al:DagSemProc.04461.3,
  author =	{Fliege, J\"{o}rg and Heermann, Christoph and Weyers, Bernd},
  title =	{{A New Adaptive Algorithm for Convex Quadratic Multicriteria Optimization}},
  booktitle =	{Practical Approaches to Multi-Objective Optimization},
  pages =	{1--39},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2005},
  volume =	{4461},
  editor =	{J\"{u}rgen Branke and Kalyanmoy Deb and Kaisa Miettinen and Ralph E. Steuer},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.04461.3},
  URN =		{urn:nbn:de:0030-drops-2386},
  doi =		{10.4230/DagSemProc.04461.3},
  annote =	{Keywords: Multicriteria optimization, warm-start methods, interior-point methods, primal-dual algorithms}
}
Document
DOI: 10.4230/DagSemProc.04461.4
A New Approach on Many Objective Diversity Measurement

Authors: Sanaz Mostaghim and Jürgen Teich

Abstract
In multi-objective particle swarm optimization (MOPSO) methods, selecting the best {it local guide} (the global best particle) for each particle of the population from a set of Pareto-optimal solutions has a great impact on the convergence and diversity of solutions, especially when optimizing problems with high number of objectives. here, we introduce the Sigma method as a new method for finding best local guides for each particle of the population. The Sigma method is implemented and is compared with another method, which uses the strategy of an existing MOPSO method for finding the local guides. These methods are examined for different test functions and the results are compared with the results of a multi-objective evolutionary algorithm (MOEA).
Cite as

Sanaz Mostaghim and Jürgen Teich. A New Approach on Many Objective Diversity Measurement. In Practical Approaches to Multi-Objective Optimization. Dagstuhl Seminar Proceedings, Volume 4461, pp. 1-15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{mostaghim_et_al:DagSemProc.04461.4,
  author =	{Mostaghim, Sanaz and Teich, J\"{u}rgen},
  title =	{{A New Approach on Many Objective Diversity Measurement}},
  booktitle =	{Practical Approaches to Multi-Objective Optimization},
  pages =	{1--15},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2005},
  volume =	{4461},
  editor =	{J\"{u}rgen Branke and Kalyanmoy Deb and Kaisa Miettinen and Ralph E. Steuer},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.04461.4},
  URN =		{urn:nbn:de:0030-drops-2543},
  doi =		{10.4230/DagSemProc.04461.4},
  annote =	{Keywords: Multi-objective Optimization, Particle Swarm Optimization}
}
Document
DOI: 10.4230/DagSemProc.04461.5
A Tutorial on Evolutionary Multi-Objective Optimization (EMO)

Authors: Kalyanmoy Deb

Abstract
Many real-world search and optimization problems are naturally posed as non-linear programming problems having multiple objectives. Due to lack of suitable solution techniques, such problems are artificially converted into a single-objective problem and solved. The difficulty arises because such problems give rise to a set of Pareto-optimal solutions, instead of a single optimum solution. It then becomes important to find not just one Pareto-optimal solution but as many of them as possible. Classical methods are not quite efficient in solving these problems because they require repetitive applications to find multiple Pareto-optimal solutions and in some occasions repetitive applications do not guarantee finding distinct Pareto-optimal solutions. The population approach of evolutionary algorithms (EAs) allows an efficient way to find multiple Pareto-optimal solutions simultaneously in a single simulation run. In this tutorial, we discussed the following aspects related to EMO: 1. The basic differences in principle of EMO with classical methods. 2. A gentle introduction to evolutionary algorithms with simple examples. A simple method of handling constraints was also discussed. 3. The concept of domination and methods of finding non-dominated solutions in a population of solutions were discussed. 4. A brief history of the development of EMO is highlighted. 5. A number of main EMO methods (NSGA-II, SPEA and PAES) were discussed. 6. The advantage of EMO methodologies was discussed by presenting a number of case studies. They clearly showed the advantage of finding a number of Pareto-optimal solutions simultaneously. 7. Three advantages of using an EMO methodology were stressed: (i) For a better decision making (in terms of choosing a compromised solution) in the presence of multiple solutions (ii) For finding important relationships among decision variables (useful in design optimization). Some case studies from engineering demonstrated the importance of such studies. (iii) For solving other optimization problems efficiently. For example, in solving genetic programming problems, the so-called `bloating problem of increased program size can be solved by using a second objective of minimizing the size of the programs. 8. A number of salient research topics were highlighted. Some of them are as follows: (i) Development of scalable test problems (ii) Development of computationally fast EMO methods (iii) Performance metrics for evaluating EMO methods (iv) Interactive EMO methodologies (v) Robust multi-objective optimization procedures (vi) Finding knee or other important solutions including partial Pareto-optimal set (vii) Multi-objective scheduling and other optimization problems. It was clear from the discussions that evolutionary search methods offers an alternate means of solving multi-objective optimization problems compared to classical approaches. This is why multi-objective optimization using EAs is getting a growing attention in the recent years. The motivated readers may explore current research issues and other important studies from various texts (Coello et al, 2003; Deb, 2001), conference proceedings (EMO-01 and EMO-03 Proceedings) and numerous research papers (http://www.lania.mx/~ccoello/EMOO/). References: ---------- C. A. C. Coello, D. A. VanVeldhuizen, and G. Lamont. Evolutionary Algorithms for Solving Multi-Objective Problems. Boston, MA: Kluwer Academic Publishers, 2002. K.Deb. Multi-objective optimization using evolutionary algorithms. Chichester, UK: Wiley, 2001. C. Fonseca, P. Fleming, E. Zitzler, K. Deb, and L. Thiele, editors. Proceedings of the Second Evolutionary Multi-Criterion Optimization (EMO-03) Conference (Lecture Notes in Computer Science (LNCS) 2632). Heidelberg: Springer, 2003. E. Zitzler, K. Deb, L. Thiele, C. A. C. Coello, and D. Corne, editors. Proceedings of the First Evolutionary Multi-Criterion Optimization (EMO-01) Conference (Lecture Notes in Computer Science (LNCS) 1993). Heidelberg: Springer, 2001.
Cite as

Kalyanmoy Deb. A Tutorial on Evolutionary Multi-Objective Optimization (EMO). In Practical Approaches to Multi-Objective Optimization. Dagstuhl Seminar Proceedings, Volume 4461, pp. 1-2, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{deb:DagSemProc.04461.5,
  author =	{Deb, Kalyanmoy},
  title =	{{A Tutorial on Evolutionary Multi-Objective Optimization (EMO)}},
  booktitle =	{Practical Approaches to Multi-Objective Optimization},
  pages =	{1--2},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2005},
  volume =	{4461},
  editor =	{J\"{u}rgen Branke and Kalyanmoy Deb and Kaisa Miettinen and Ralph E. Steuer},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.04461.5},
  URN =		{urn:nbn:de:0030-drops-2520},
  doi =		{10.4230/DagSemProc.04461.5},
  annote =	{Keywords: Multi-objective optimization, multi-criterion optimization, Pareto-optimal solutions, Evolutionary methods, EMO}
}
Document
DOI: 10.4230/DagSemProc.04461.6
An Adaptive Scheme to Generate the Pareto Front Based on the Epsilon-Constraint Method

Authors: Marco Laumanns, Lothar Thiele, and Eckart Zitzler

Abstract
We discuss methods for generating or approximating the Pareto set of multiobjective optimization problems by solving a sequence of constrained single-objective problems. The necessity of determining the constraint value a priori is shown to be a serious drawback of the original epsilon-constraint method. We therefore propose a new, adaptive scheme to generate appropriate constraint values during the run. A simple example problem is presented, where the running time (measured by the number of constrained single-objective sub-problems to be solved) of the original epsilon-constraint method is exponential in the problem size (number of decision variables), although the size of the Pareto set grows only linearly. We prove that --- independent of the problem or the problem size --- the time complexity of the new scheme is O(k^{m-1}), where k is the number of Pareto-optimal solutions to be found and m the number of objectives. Simulation results for the example problem as well as for different instances of the multiobjective knapsack problem demonstrate the behavior of the method, and links to reference implementations are provided.
Cite as

Marco Laumanns, Lothar Thiele, and Eckart Zitzler. An Adaptive Scheme to Generate the Pareto Front Based on the Epsilon-Constraint Method. In Practical Approaches to Multi-Objective Optimization. Dagstuhl Seminar Proceedings, Volume 4461, pp. 1-11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{laumanns_et_al:DagSemProc.04461.6,
  author =	{Laumanns, Marco and Thiele, Lothar and Zitzler, Eckart},
  title =	{{An Adaptive Scheme to Generate the Pareto Front Based on the Epsilon-Constraint Method}},
  booktitle =	{Practical Approaches to Multi-Objective Optimization},
  pages =	{1--11},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2005},
  volume =	{4461},
  editor =	{J\"{u}rgen Branke and Kalyanmoy Deb and Kaisa Miettinen and Ralph E. Steuer},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.04461.6},
  URN =		{urn:nbn:de:0030-drops-2465},
  doi =		{10.4230/DagSemProc.04461.6},
  annote =	{Keywords: Multiple objective optimization, non-dominated set, Pareto set, epsilon-constraint method, generating methods}
}
Document
DOI: 10.4230/DagSemProc.04461.7
Application Issues for Multiobjective Evolutionary Algorithms

Authors: Thomas Hanne

Abstract
In the talk, various issues of the design and application of multiobjective evolutionary algorithms for real-life optimization problems are discussed. In particular, questions on problem-specific data structures and evolutionary operators and the determination of method parameters are treated. Three application examples in the areas of constrained global optimization (electronic circuit design), semi-infinite programming (design centering problems), and discrete optimization (project scheduling) are discussed.
Cite as

Thomas Hanne. Application Issues for Multiobjective Evolutionary Algorithms. In Practical Approaches to Multi-Objective Optimization. Dagstuhl Seminar Proceedings, Volume 4461, pp. 1-11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{hanne:DagSemProc.04461.7,
  author =	{Hanne, Thomas},
  title =	{{Application Issues for Multiobjective Evolutionary Algorithms}},
  booktitle =	{Practical Approaches to Multi-Objective Optimization},
  pages =	{1--11},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2005},
  volume =	{4461},
  editor =	{J\"{u}rgen Branke and Kalyanmoy Deb and Kaisa Miettinen and Ralph E. Steuer},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.04461.7},
  URN =		{urn:nbn:de:0030-drops-3443},
  doi =		{10.4230/DagSemProc.04461.7},
  annote =	{Keywords: Multiobjective optimization, Pareto set, evolutionary algorithm, discrete optimization, continuous optimization, electronic circuit design, semi-infin}
}
Document
DOI: 10.4230/DagSemProc.04461.8
Approximation and Visualization of Pareto Frontier in the Framework of Classical Approach to Multi-Objective Optimization

Authors: Alexander Lotov

Abstract
This paper is devoted to a Pareto frontier generation technique, which is aimed at subsequent visualization of the Pareto frontier in an interaction with the user. This technique known as the Interactive Decision Maps technique was initiated about 30 years ago. Now it is applied for decision support in both convex and non-convex decision problems in various fields, from machinery design to environmental planning. The number of conflicting criteria explored with the help of the Interactive Decision Maps technique is usually between three and seven, but some users manage to apply the technique in the case of a larger number of criteria. Here we outline the main ideas of the technique, concentrating at nonlinear problems.
Cite as

Alexander Lotov. Approximation and Visualization of Pareto Frontier in the Framework of Classical Approach to Multi-Objective Optimization. In Practical Approaches to Multi-Objective Optimization. Dagstuhl Seminar Proceedings, Volume 4461, pp. 1-17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{lotov:DagSemProc.04461.8,
  author =	{Lotov, Alexander},
  title =	{{Approximation and Visualization of Pareto Frontier in the Framework of Classical Approach to Multi-Objective Optimization}},
  booktitle =	{Practical Approaches to Multi-Objective Optimization},
  pages =	{1--17},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2005},
  volume =	{4461},
  editor =	{J\"{u}rgen Branke and Kalyanmoy Deb and Kaisa Miettinen and Ralph E. Steuer},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.04461.8},
  URN =		{urn:nbn:de:0030-drops-2357},
  doi =		{10.4230/DagSemProc.04461.8},
  annote =	{Keywords: Multi-objective optimization, Pareto frontier, visualization}
}
Document
DOI: 10.4230/DagSemProc.04461.9
Current Status of the EMOO Repository, Including Current and Future Research Trends

Authors: Carlos A. Coello Coello

Abstract
In this talk, Ill present some statistics of the EMOO repository (delta.cs.cinvestav.mx/~ccoello/EMOO/), emphasizing some of the trends that have been detected in terms of basic research and applications of multi-objective evolutionary algorithms. For example, Ill discuss the remarkable increase in PhD theses related to EMOO, as well as the number of journal papers and exposure of the area in evolutionary computation conferences. Finally, some (potential) future research trends will also be discussed.
Cite as

Carlos A. Coello Coello. Current Status of the EMOO Repository, Including Current and Future Research Trends. In Practical Approaches to Multi-Objective Optimization. Dagstuhl Seminar Proceedings, Volume 4461, pp. 1-5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{coellocoello:DagSemProc.04461.9,
  author =	{Coello Coello, Carlos A.},
  title =	{{Current Status of the EMOO Repository, Including Current and Future Research Trends}},
  booktitle =	{Practical Approaches to Multi-Objective Optimization},
  pages =	{1--5},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2005},
  volume =	{4461},
  editor =	{J\"{u}rgen Branke and Kalyanmoy Deb and Kaisa Miettinen and Ralph E. Steuer},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.04461.9},
  URN =		{urn:nbn:de:0030-drops-2505},
  doi =		{10.4230/DagSemProc.04461.9},
  annote =	{Keywords: Evolutionary multiobjective optimization, multiobjective optimization, repository}
}
Document
DOI: 10.4230/DagSemProc.04461.10
Effects of Crossover Operations on the Performance of EMO Algorithms

Authors: Hisao Ishibuchi

Abstract
This paper visually demonstrates the effect of crossover operations on the performance of EMO algorithms through computational experiments on multi-objective 0/1 knapsack problems. In our computational experiments, we use the NSGA-II algorithm as a representative EMO algorithm. First we compare the performance of the NSGA-II algorithm between two cases: NSGA-II with/without crossover. Experimental results show that the crossover operation has a positive effect on the convergence of solutions to the Pareto front and a negative effect on the diversity of solutions. That is, the crossover operation decreases the diversity of solutions while it improves the convergence of solutions to the Pareto front. Next we examine the effects of recombining similar or dissimilar parents using a similarity-based mating scheme. Experimental results show that the performance of the NSGA-II algorithm is improved by recombining similar parents and degraded by recombining dissimilar ones. Finally we show that the recombination of extreme and similar parents using the similarity-based mating scheme drastically improves the diversity of obtained non-dominated solutions without severely degrading their convergence to the Pareto front. An idea of dynamically controlling the selection pressure toward extreme and similar parents is also illustrated through computational experiments.
Cite as

Hisao Ishibuchi. Effects of Crossover Operations on the Performance of EMO Algorithms. In Practical Approaches to Multi-Objective Optimization. Dagstuhl Seminar Proceedings, Volume 4461, pp. 1-8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{ishibuchi:DagSemProc.04461.10,
  author =	{Ishibuchi, Hisao},
  title =	{{Effects of Crossover Operations on the Performance of EMO Algorithms}},
  booktitle =	{Practical Approaches to Multi-Objective Optimization},
  pages =	{1--8},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2005},
  volume =	{4461},
  editor =	{J\"{u}rgen Branke and Kalyanmoy Deb and Kaisa Miettinen and Ralph E. Steuer},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.04461.10},
  URN =		{urn:nbn:de:0030-drops-2394},
  doi =		{10.4230/DagSemProc.04461.10},
  annote =	{Keywords: Evolutionary Multiobjective Optimization, Multiobjective 0/1 Knapsack Problems, Crossover Operations, Mating Restriction}
}
Document
DOI: 10.4230/DagSemProc.04461.11
Hybrid Representations for Composition Optimization and Parallelizing MOEAs

Authors: Felix Streichert, Holger Ulmer, and Andreas Zell

Abstract
We present a hybrid EA representation suitable to optimize composition optimization problems ranging from optimizing recipes for catalytic materials to cardinality constrained portfolio selection. On several problem instances we can show that this new representation performs better than standard repair mechanisms with Lamarckism. Additionally, we investigate the a clustering based parallelization scheme for MOEAs. We prove that typical "divide and conquer'' approaches are not suitable for the standard test functions like ZDT 1-6. Therefore, we suggest a new test function based on the portfolio selection problem and prove the feasibility of "divide and conquer'' approaches on this test function.
Cite as

Felix Streichert, Holger Ulmer, and Andreas Zell. Hybrid Representations for Composition Optimization and Parallelizing MOEAs. In Practical Approaches to Multi-Objective Optimization. Dagstuhl Seminar Proceedings, Volume 4461, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{streichert_et_al:DagSemProc.04461.11,
  author =	{Streichert, Felix and Ulmer, Holger and Zell, Andreas},
  title =	{{Hybrid Representations for Composition Optimization and Parallelizing MOEAs}},
  booktitle =	{Practical Approaches to Multi-Objective Optimization},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2005},
  volume =	{4461},
  editor =	{J\"{u}rgen Branke and Kalyanmoy Deb and Kaisa Miettinen and Ralph E. Steuer},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.04461.11},
  URN =		{urn:nbn:de:0030-drops-2519},
  doi =		{10.4230/DagSemProc.04461.11},
  annote =	{Keywords: Multi-objective Evolutionary Algorithms (MOEAs), Solution Representation, Constrained Portfolio Selection Problem, Parallelizing MOEAs}
}
Document
DOI: 10.4230/DagSemProc.04461.12
Multi-criteria ranking of a finite set of alternatives using ordinal regression and additive utility functions - a new UTA-GMS method

Authors: Roman Slowinski, Salvatore Greco, and Vincent Mousseau

Abstract
UTA-GMS is a new method for assessment of strong or weak outranking relation in a problem of multi-criteria ranking, proposed by the authors. The ranking concerns a finite but relatively large set of alternatives A. We assume indirect preference information supplied by the decision maker (DM) in form of a complete preorder on a subset of reference alternatives R, called reference preorder. The preference model build from this information is an additive value function. The technique of passing from reference preorder to compatible additive value functions is called ordinal regression and it is well known from the UTA method proposed by Jacquet-Lagreze and Siskos in 1982. Unlike in the UTA method, we take into account all compatible value functions (instead of one or several most characteristic) at the stage of ranking the whole set A of alternatives. Moreover, we do not impose the additive value function to have piecewise-linear components but we accept any additive form. The resulting relations in A are twofold: strong outranking (if alternative x has greater value than y for all compatible value functions) and weak outranking (if alternative x has greater value than y for at least one compatible value function). Strong outranking is a partial preorder and weak outranking is a complete preorder in A. The strong outranking is of particular interest for the DM – it corresponds to dominance relation when the set of reference alternatives is empty, and to a complete preorder relation when the reference ranking is compatible with a single value function only. This approach has several interesting extensions useful for practical applications. The method has been implemented for a PC and will be presented together with an example of application.
Cite as

Roman Slowinski, Salvatore Greco, and Vincent Mousseau. Multi-criteria ranking of a finite set of alternatives using ordinal regression and additive utility functions - a new UTA-GMS method. In Practical Approaches to Multi-Objective Optimization. Dagstuhl Seminar Proceedings, Volume 4461, pp. 1-4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{slowinski_et_al:DagSemProc.04461.12,
  author =	{Slowinski, Roman and Greco, Salvatore and Mousseau, Vincent},
  title =	{{Multi-criteria ranking of a finite set of alternatives using ordinal regression and additive utility functions - a new UTA-GMS method}},
  booktitle =	{Practical Approaches to Multi-Objective Optimization},
  pages =	{1--4},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2005},
  volume =	{4461},
  editor =	{J\"{u}rgen Branke and Kalyanmoy Deb and Kaisa Miettinen and Ralph E. Steuer},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.04461.12},
  URN =		{urn:nbn:de:0030-drops-2476},
  doi =		{10.4230/DagSemProc.04461.12},
  annote =	{Keywords: Multiple-criteria ranking, ordinal regression, partial preorder, UTA-like method}
}
Document
DOI: 10.4230/DagSemProc.04461.13
Multi-objective Optimization and its Engineering Applications

Authors: Hirotaka Nakayama

Abstract
Many practical optimization problems usually have several conflicting objectives. In those multi-objective optimization, no solution optimizing all objective functions simultaneously exists in general. Instead, Pareto optimal solutions, which are ``efficient" in terms of all objective functions, are introduced. In general we have many Pareto optimal solutions. Therefore, we need to decide a final solution among Pareto optimal solutions taking into account the balance among objective functions, which is called ``trade-off analysis". It is no exaggeration to say that the most important task in multi-objective optimization is trade-off analysis. Consequently, the methodology should be discussed in view of how it is easy and understandable for trade-off analysis. In cases with two or three objective functions, the set of Pareto optimal solutions in the objective function space (i.e., Pareto frontier) can be depicted relatively easily. Seeing Pareto frontiers, we can grasp the trade-off relation among objectives totally. Therefore, it would be the best way to depict Pareto frontiers in cases with two or three objectives. (It might be difficult to read the trade-off relation among objectives with three dimension, though). In cases with more than three objectives, however, it is impossible to depict Pareto forntier. Under this circumstance, interactive methods can help us to make local trade-off analysis showing a ``certain" Pareto optimal solution. A number of methods differing in which Pareto optimal solution is to be shown, have been developed. This paper discusses critical issues among those methods for multi-objective optimization, in particular applied to engineering design problems.
Cite as

Hirotaka Nakayama. Multi-objective Optimization and its Engineering Applications. In Practical Approaches to Multi-Objective Optimization. Dagstuhl Seminar Proceedings, Volume 4461, pp. 1-13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{nakayama:DagSemProc.04461.13,
  author =	{Nakayama, Hirotaka},
  title =	{{Multi-objective Optimization and its Engineering Applications}},
  booktitle =	{Practical Approaches to Multi-Objective Optimization},
  pages =	{1--13},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2005},
  volume =	{4461},
  editor =	{J\"{u}rgen Branke and Kalyanmoy Deb and Kaisa Miettinen and Ralph E. Steuer},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.04461.13},
  URN =		{urn:nbn:de:0030-drops-2346},
  doi =		{10.4230/DagSemProc.04461.13},
  annote =	{Keywords: Multi-Objective Optimization, Interactive Multi-Objective Optimization, Evolutionary Algorithms, Pareto Frontier}
}
Document
DOI: 10.4230/DagSemProc.04461.14
Multiobjective Optimization and Multiple Constraint Handling with Evolutionary Algorithms

Authors: Carlos M. Fonseca and Peter J. Fleming

Abstract
In this talk, fitness assignment in multiobjective evolutionary algorithms is interpreted as a multi-criterion decision process. A suitable decision making framework based on goals and priorities is formulated in terms of a relational operator, characterized, and shown to encompass a number of simpler decision strategies, including constraint satisfaction, lexicographic optimization, and a form of goal programming. Then, the ranking of an arbitrary number of candidates is considered, and the effect of preference changes on the cost surface seen by an evolutionary algorithm is illustrated graphically for a simple problem. The formulation of a multiobjective genetic algorithm based on the proposed decision strategy is also discussed. Niche formation techniques are used to promote diversity among preferable candidates, and progressive articulation of preferences is shown to be possible as long as the genetic algorithm can recover from abrupt changes in the cost landscape. Finally, an application to the optimization of the low-pressure spool speed governor of a Pegasus gas turbine engine is described, which illustrates how a technique such as the Multiobjective Genetic Algorithm can be applied, and exemplifies how design requirements can be refined as the algorithm runs. The two instances of the problem studied demonstrate the need for preference articulation in cases where many and highly competing objectives lead to a non-dominated set too large for a finite population to sample effectively. It is shown that only a very small portion of the non-dominated set is of practical relevance, which further substantiates the need to supply preference information to the GA.
Cite as

Carlos M. Fonseca and Peter J. Fleming. Multiobjective Optimization and Multiple Constraint Handling with Evolutionary Algorithms. In Practical Approaches to Multi-Objective Optimization. Dagstuhl Seminar Proceedings, Volume 4461, pp. 1-2, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{fonseca_et_al:DagSemProc.04461.14,
  author =	{Fonseca, Carlos M. and Fleming, Peter J.},
  title =	{{Multiobjective Optimization and Multiple Constraint Handling with Evolutionary Algorithms}},
  booktitle =	{Practical Approaches to Multi-Objective Optimization},
  pages =	{1--2},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2005},
  volume =	{4461},
  editor =	{J\"{u}rgen Branke and Kalyanmoy Deb and Kaisa Miettinen and Ralph E. Steuer},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.04461.14},
  URN =		{urn:nbn:de:0030-drops-2371},
  doi =		{10.4230/DagSemProc.04461.14},
  annote =	{Keywords: Evolutionary algorithms, multiobjective optimization, preference articulation, interactive optimization.}
}
Document
DOI: 10.4230/DagSemProc.04461.15
NBI and MOGA-II, two complementary algorithms for Multi-Objective optimizations

Authors: Enrico Rigoni and Silvia Poles

Abstract
The NBI-NLPQLP optimization method is tested on several multi-objective optimization problems. Its performance is compared to that of MOGA-II: since NBI-NLPQLP is based on the classical gradient-based NLPQLP, it is fast and accurate, but not as robust, in comparison with the genetic algorithm. Furthermore a discontinuous Pareto frontier can give rise to problems in the NBI´s convergence. In order to overcome this problem, a hybridization technique coupled with a partitioning method is proposed.
Cite as

Enrico Rigoni and Silvia Poles. NBI and MOGA-II, two complementary algorithms for Multi-Objective optimizations. In Practical Approaches to Multi-Objective Optimization. Dagstuhl Seminar Proceedings, Volume 4461, pp. 1-22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{rigoni_et_al:DagSemProc.04461.15,
  author =	{Rigoni, Enrico and Poles, Silvia},
  title =	{{NBI and MOGA-II, two complementary algorithms for Multi-Objective optimizations}},
  booktitle =	{Practical Approaches to Multi-Objective Optimization},
  pages =	{1--22},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2005},
  volume =	{4461},
  editor =	{J\"{u}rgen Branke and Kalyanmoy Deb and Kaisa Miettinen and Ralph E. Steuer},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.04461.15},
  URN =		{urn:nbn:de:0030-drops-2728},
  doi =		{10.4230/DagSemProc.04461.15},
  annote =	{Keywords: Genetic Algorithms, Normal-Boundary Intersection, Designs optimizations}
}
Document
DOI: 10.4230/DagSemProc.04461.16
On Continuation Methods for the Numerical Treatment of Multi-Objective Optimization Problems

Authors: Oliver Schütze, Alessandro Dell'Aere, and Michael Dellnitz

Abstract
In this report we describe how continuation methods can be used for the numerical treatment of multi-objective optimization problems (MOPs): starting with a given Karush-Kuhn-Tucker point (KKT-point) x of an MOP, these techniques can be applied to detect further KKT-points in the neighborhood of x. In the next step, again further points are computed starting with these new-found KKT-points, and so on. In order to maintain a good spread of these solutions we use boxes for the representation of the computed parts of the solution set. Based on this background, we propose a new predictor-corrector variant, and show some numerical results indicating the strength of the method, in particular in higher dimensions. Further, the data structure allows for an efficient computation of MOPs with more than two objectives, which has not been considered so far in most existing continuation methods.
Cite as

Oliver Schütze, Alessandro Dell'Aere, and Michael Dellnitz. On Continuation Methods for the Numerical Treatment of Multi-Objective Optimization Problems. In Practical Approaches to Multi-Objective Optimization. Dagstuhl Seminar Proceedings, Volume 4461, pp. 1-15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{schutze_et_al:DagSemProc.04461.16,
  author =	{Sch\"{u}tze, Oliver and Dell'Aere, Alessandro and Dellnitz, Michael},
  title =	{{On Continuation Methods for the Numerical Treatment of Multi-Objective Optimization Problems}},
  booktitle =	{Practical Approaches to Multi-Objective Optimization},
  pages =	{1--15},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2005},
  volume =	{4461},
  editor =	{J\"{u}rgen Branke and Kalyanmoy Deb and Kaisa Miettinen and Ralph E. Steuer},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.04461.16},
  URN =		{urn:nbn:de:0030-drops-3497},
  doi =		{10.4230/DagSemProc.04461.16},
  annote =	{Keywords: multi-objective optimization, continuation, k-manifolds}
}
Document
DOI: 10.4230/DagSemProc.04461.17
On Properly Pareto Optimal Solutions

Authors: Pradyumn Kumar Shukla, Joydeep Dutta, and Kalyanmoy Deb

Abstract
In this paper we study epsilon-proper efficiency in multiobjective optimization. We introduce various new definitions of epsilon-proper efficiency, relate them with existing ones, study various concepts and develop very general necessary optimality conditions for a few of them.
Cite as

Pradyumn Kumar Shukla, Joydeep Dutta, and Kalyanmoy Deb. On Properly Pareto Optimal Solutions. In Practical Approaches to Multi-Objective Optimization. Dagstuhl Seminar Proceedings, Volume 4461, pp. 1-5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{shukla_et_al:DagSemProc.04461.17,
  author =	{Shukla, Pradyumn Kumar and Dutta, Joydeep and Deb, Kalyanmoy},
  title =	{{On Properly Pareto Optimal Solutions}},
  booktitle =	{Practical Approaches to Multi-Objective Optimization},
  pages =	{1--5},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2005},
  volume =	{4461},
  editor =	{J\"{u}rgen Branke and Kalyanmoy Deb and Kaisa Miettinen and Ralph E. Steuer},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.04461.17},
  URN =		{urn:nbn:de:0030-drops-2403},
  doi =		{10.4230/DagSemProc.04461.17},
  annote =	{Keywords: Proper efficiency, epsilon solutions}
}
Document
DOI: 10.4230/DagSemProc.04461.18
Optimizing Surface Profiles during Hot Rolling: A Genetic Algorithms based Multi-objective Analysis

Authors: Nirupam Chakraborti, Barrenkala Siva Kumar, Satish V. Babu, Sri Subhrangshu Moitra, and Ananya Mukhopadhyay

Abstract
A hot rolled strip produced by any integrated steel plant would require satisfying some stringent requirements of its surface profile. Crown and Flatness are two industrially accepted quantifiers that relate to the geometric tolerances in the rolled strips. This study attempts to regulate both crown and flatness within an acceptable limit, satisfying more than one objective at a time. Mathematically, this leads to a multi-objective optimization problem where the solution is no longer unique and a family of equally feasible solutions leads to the so called Pareto-Front, where each member is simply as good as the others. To implement this concept in the present context, one needs to realize that the surface deformation, which is ultimately imparted to the rolled sheets, comes from more than one source. The wear of the rolls, their thermal expansion, bending, and also deformation, contribute significantly towards the crown and flatness that is ultimately observed. During this study a detailed mathematical model has been worked out for this process incorporating all of these phenomena. Computation for the Pareto-optimality has been carried out using different forms of biologically inspired Genetic Algorithms, often integrated with an Ant Colony Optimization Scheme. Ultimately the model has been fine tuned for the hot rolling practice in a major integrated steel plant and tested against their actual operational data.
Cite as

Nirupam Chakraborti, Barrenkala Siva Kumar, Satish V. Babu, Sri Subhrangshu Moitra, and Ananya Mukhopadhyay. Optimizing Surface Profiles during Hot Rolling: A Genetic Algorithms based Multi-objective Analysis. In Practical Approaches to Multi-Objective Optimization. Dagstuhl Seminar Proceedings, Volume 4461, pp. 1-12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{chakraborti_et_al:DagSemProc.04461.18,
  author =	{Chakraborti, Nirupam and Kumar, Barrenkala Siva and Babu, Satish V. and Moitra, Sri Subhrangshu and Mukhopadhyay, Ananya},
  title =	{{Optimizing Surface Profiles during Hot Rolling: A Genetic Algorithms based Multi-objective Analysis}},
  booktitle =	{Practical Approaches to Multi-Objective Optimization},
  pages =	{1--12},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2005},
  volume =	{4461},
  editor =	{J\"{u}rgen Branke and Kalyanmoy Deb and Kaisa Miettinen and Ralph E. Steuer},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.04461.18},
  URN =		{urn:nbn:de:0030-drops-2454},
  doi =		{10.4230/DagSemProc.04461.18},
  annote =	{Keywords: Rolling, Hot Rolling, Crown, Flatness, Genetic Algorithms, Ant Colony Optimization, Multi-objective Optimization, Pareto Front, Multi-objective Evolut}
}
Document
DOI: 10.4230/DagSemProc.04461.19
Runtime Analysis of a Simple Multi-Objective Evolutionary Algorithm

Authors: Oliver Giel

Abstract
Practical knowledge on the design and application of multi-objective evolutionary algorithms (MOEAs) is available but well-founded theoretical analyses of the runtime are rare. Laumanns, Thiele, Zitzler, Welzel and Deb (2002) have started such an analysis for two simple mutation-based algorithms including SEMO. These algorithms search locally in the neighborhood of their current population by selecting an individual and flipping one randomly chosen bit. Due to its local search operator, SEMO cannot escape from local optima, and, therefore, has no finite expected runtime in general. In this talk, we investigate the runtime of a variant of SEMO whose mutation operator flips each bit independently. It is proven that its expected runtime is O(n^n) for all objective functions f: {0,1}^n -> R^m, and that there are bicriteria problems among the hardest problem for this algorithm. Moreover, for each d between 2 and n, a bicriteria problem with expected runtime Theta(n^d) is presented. This shows that bicriteria problems cover the full range of potential runtimes of this variant of SEMO. For the problem LOTZ (Leading-Ones-Trailing Zeroes), the runtime does not increase substantially if we use the global search operator. Finally, we consider the problem MOCO (Multi-Objective-Counting-Ones). We show that the conjectured bound O((n^2)log n) on the expected runtime is wrong for both variants of SEMO. In fact, MOCO is almost a worst case example for SEMO if we consider the expected runtime; however, the runtime is O((n^2)log n) with high probability. Some ideas from the proof will be presented.
Cite as

Oliver Giel. Runtime Analysis of a Simple Multi-Objective Evolutionary Algorithm. In Practical Approaches to Multi-Objective Optimization. Dagstuhl Seminar Proceedings, Volume 4461, pp. 1-4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{giel:DagSemProc.04461.19,
  author =	{Giel, Oliver},
  title =	{{Runtime Analysis of a Simple Multi-Objective Evolutionary Algorithm}},
  booktitle =	{Practical Approaches to Multi-Objective Optimization},
  pages =	{1--4},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2005},
  volume =	{4461},
  editor =	{J\"{u}rgen Branke and Kalyanmoy Deb and Kaisa Miettinen and Ralph E. Steuer},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.04461.19},
  URN =		{urn:nbn:de:0030-drops-2711},
  doi =		{10.4230/DagSemProc.04461.19},
  annote =	{Keywords: Runtime analysis, multi-objecive evolutionary algorithms}
}
Document
DOI: 10.4230/DagSemProc.04461.20
Visualization of Global Trade-Offs in Aerodynamic Problems by ARMOGAs

Authors: Daisuke Sasaki and Shigeru Obayashi

Abstract
Trade-offs is one of important elements for engineering design problems characterized by multiple conflicting design objectives to be simultaneously improved. In many design problems such as aerodynamic design, due to computational reasons, only a limited number of evaluations can be allowed for industrial use. Efficient MOEAs, Adaptive Range Multi-Objective Genetic Algorithms (ARMOGAs), to identify trade-offs using a small number of function evaluations have been developed. In this study, ARMOGAs are applied to aerodynamic designs problems to identify trade-offs efficiently. In addition to identify trade-offs, trade-off analysis is also important to obtain useful knowledge about the design problem. To analyze the high-dimensional data of aerodynamic optimization problem, Self-Organizing Maps are applied to understand the trade-offs.
Cite as

Daisuke Sasaki and Shigeru Obayashi. Visualization of Global Trade-Offs in Aerodynamic Problems by ARMOGAs. In Practical Approaches to Multi-Objective Optimization. Dagstuhl Seminar Proceedings, Volume 4461, pp. 1-14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{sasaki_et_al:DagSemProc.04461.20,
  author =	{Sasaki, Daisuke and Obayashi, Shigeru},
  title =	{{Visualization of Global Trade-Offs in Aerodynamic Problems by ARMOGAs}},
  booktitle =	{Practical Approaches to Multi-Objective Optimization},
  pages =	{1--14},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2005},
  volume =	{4461},
  editor =	{J\"{u}rgen Branke and Kalyanmoy Deb and Kaisa Miettinen and Ralph E. Steuer},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.04461.20},
  URN =		{urn:nbn:de:0030-drops-2418},
  doi =		{10.4230/DagSemProc.04461.20},
  annote =	{Keywords: Aerodynamic optimization, MOEA, SOM, trade-off analysis}
}

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  • Dell'Aere, Alessandro
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  • Dutta, Joydeep
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  • Fonseca, Carlos M.
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